# nLab horizontal categorification

Contents

category theory

## Applications

#### Algebra

higher algebra

universal algebra

categorification

# Contents

## Idea

Horizontal categorification or Oidification describes the process by which

1. a concept is realized to be equivalent to a certain type of category or magmoid with a single object;

2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.

## Remarks

• This is to be contrasted with vertical categorification.

• It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.

• It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.

## Further discussion

Related $n$-Café-discussion is in

Last revised on February 7, 2023 at 13:08:37. See the history of this page for a list of all contributions to it.